In radian measure, the angles of a 45-45-90 triangle are:
A. $\frac{\pi}{8}, \frac{\pi}{8}, \frac{\pi}{2}$
B. $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}$
C. $\frac{\pi}{3}, \frac{\pi}{3}, \frac{\pi}{2}$
D. $\frac{\pi}{6}, \frac{\pi}{6}, \frac{\pi}{2}$
Answer (1)
Convert the given angles in degrees to radians using the conversion factor 180 π .
Calculate the radian measure of 45 degrees: 45 × 180 π = 4 π .
Calculate the radian measure of 90 degrees: 90 × 180 π = 2 π .
The angles in radians are 4 π , 4 π , 2 π .
Explanation
Understanding the problem A 45-45-90 triangle has angles that measure 45 degrees, 45 degrees, and 90 degrees. To express these angles in radians, we need to convert from degrees to radians.
Converting to radians To convert degrees to radians, we use the conversion factor 180 π .
For 45 degrees: 45 × 180 π = 180 45 π = 4 π
For 90 degrees: 90 × 180 π = 180 90 π = 2 π
Identifying the correct option Therefore, the angles in radians are 4 π , 4 π , and 2 π . Comparing these values with the given options, we find that option B matches our result.
Final Answer The angles of a 45-45-90 triangle in radian measure are 4 π , 4 π , 2 π .
Examples
Understanding angles in radians is crucial in many fields, such as physics and engineering. For example, when analyzing circular motion, radians provide a natural and convenient way to measure angles, simplifying calculations of angular velocity and acceleration. Imagine designing a rotating fan; using radians helps you accurately determine the speed and position of the blades, ensuring optimal performance and safety.
